BOOKCOMP, Inc. — John Wiley & Sons / Page 482 / 2nd Proofs / Heat Transfer Handbook / Bejan 482 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [482], (44) Lines: 2114 to 2165 ——— 2.71744pt PgVar ——— Normal Page PgEnds: T E X [482], (44) T + − δT + 0 = y + 0 dy + H /ν (6.149) where the second term on the left is the temperature difference across which heat is transferred by conduction. With H ν = 1 Pr T ν = κ Pr T y + + δy + 0 (6.150a) then T + = δT + 0 + Pr T κ ln 32.6y + Re k (6.150b) A local heat transfer coefficient can be defined based on δT 0 : δT + 0 = ρc p v ∗ h k = 1 St k (6.151) Based on experimental data for roughness from spheres, St k = C · Re −0.20 k · Pr −0.40 (6.152) with C ≈ 0.8 and Pr T ≈ 0.9, the law of the wall can be rewritten as T + = 1 St k + Pr T κ ln 32.6y + Re k (6.153) and using a procedure similar to that used for a smooth surface, St = C f /2 Pr T + (C f /2) 1/2 /St k (6.154) 6.4.16 Some Empirical Transport Correlations Cylinder in Crossflow The analytical solutions described in Sections 6.4.9 and 6.4.10 provide local convection coefficients from the front stagnation point (using similarity theory) to the separation point of a cylinder using the Smith–Spalding method. At the front stagnation point, the free stream is brought to rest, with an accompanying rise in pressure. The initial development of the boundary layer along the cylinder following this point is under favorable pressure gradient conditions; that is, dp/dx < 0. However, the pressure reaches a minimum at some value of x, depending on the Reynolds number. Farther downstream, the pressure gradient is adverse (i.e., dp/dx > 0), until the point of boundary layer separation where the surface shear stress becomes zero. This results in the formation of a wake. For Re D ≤ 2 ×10 5 , the boundary layer remains laminar until the separation point, which BOOKCOMP, Inc. — John Wiley & Sons / Page 483 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM ARRAYS OF OBJECTS 483 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [483], (45) Lines: 2165 to 2208 ——— 1.36206pt PgVar ——— Normal Page PgEnds: T E X [483], (45) occurs at an arc of about 81°. For higher values of Re D , the boundary layer undergoes transition to turbulence, which results in higher fluid momentum and the pushing back of the separation point to about 140°. The foregoing changes result in large variations in the transport behavior with the angle θ. If the interest is on an average heat transfer coefficient, many empirical equations are available. One such correlation for an isothermal cylinder that is based on the data of many investigators for various Re D and Pr, such that Re D · Pr > 0.2 was proposed by Churchill and Bernstein (1977): Nu D ≡ ¯ hD k = 0.30 + 0.62Re 1/2 D · Pr 1/3 [1 + (0.40/Pr) 2/3 ] 1/4 1 + Re D 282,000 5/8 4/5 (6.155) The fluid properties are evaluated at the film temperature, which is the average of the surface and ambient fluid values. Flow Over an Isothermal Sphere Whitaker (1972) recommends thecorrelation Nu D = 2 + 0.4Re 1/2 D + 0.06Re 2/3 D Pr 0.4 µ µ s 1/4 (6.156) The uncertainty band around this correlation is ±30% in the range: 0.71 < Pr 380, 3.5 < Re D < 7.6 × 10 4 , and 1.0 < µ/µ s < 3.2. All properties except µ s are evaluated at the ambient temperature. 6.5 HEAT TRANSFER FROM ARRAYS OF OBJECTS 6.5.1 Crossflow across Tube Banks This configuration is encountered in many practical heat transfer applications, includ- ing shell-and-tube heat exchangers. Zhukauskas (1972, 1987) has provided compre- hensive heat transfer and pressure drop correlations for tube banks in aligned and staggered configurations (Fig. 6.15). The heat transfer from a tube depends on its lo- cation within the bank. In the range of lower Reynolds numbers, typically the tubes in the first row show similar heat transfer to those in the inner rows. For higher Reynolds numbers, flow turbulence leads to higher heat transfer from inner tubes than from the first row. The heat transfer becomes invariant with tube location following the third or fourth row in the mixed-flow regime, occurring above Re D,max . The average Nusselt number from a tube is then of the form Nu D = C · Re m D,max · Pr 0.36 Pr Pr s 1/4 (6.157) for 0.7 ≤ Pr < 5000, 1 < Re D,max < 2 ×10 4 , and N L ≥ 20 and where m is as given in Table 6.2. The Reynolds number Re D,max in eq. (6.157) is based on the maximum BOOKCOMP, Inc. — John Wiley & Sons / Page 484 / 2nd Proofs / Heat Transfer Handbook / Bejan 484 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [484], (46) Lines: 2208 to 2271 ——— -3.1679pt PgVar ——— Long Page PgEnds: T E X [484], (46) V, T ϱ V, T ϱ S T S T S D S L S L A 1 A 1 A 2 A 2 ( ) Aligned tube banka ( ) Staggered tube bankb D D Figure 6.15 Tube bundle configurations studied by Zhukaukas (1987). fluid velocity within the tube bank which occurs at the mimimum-flow cross section. For the aligned arrangement, it occurs at A 1 in Fig. 6.15 and is given by V max = S T S T − D V (6.158) where V is the upstream fluid velocity. For the staggered configuration, the maximum fluid velocity occurs at A 2 if 2(S D − D)<(S T − D), in which case V max = S T 2(S D − D) V (6.159) Otherwise, the aligned tube expression of eq. (6.158) can be used. TABLE 6.2 Parameters Re D,max ,C,andm for Various Aligned and Staggered Tube Arrangements Configuration Re D,max Cm Aligned 16–100 0.90 0.40 Staggered 1.6–40 1.04 0.40 Aligned 100–1000 0.52 0.50 Staggered 40–1000 0.71 0.50 Aligned (S T /S L > 0.7) 1000–2 × 10 5 0.27 0.63 Staggered (S T /S L < 2) 1000–2 × 10 5 0.35 (S T /S L ) 0.20 0.60 Staggered (S T /S L > 2) 1000–2 × 10 5 0.40 0.60 Aligned 2 ×10 5 –2 ×10 6 0.033 0.80 Staggered (Pr > 0.70) 2 ×10 5 –2 ×10 6 0.031 (S T /S L ) 0.20 0.80 Staggered (Pr = 0.70) 2 ×10 5 –2 ×10 6 0.027 (S T /S L ) 0.20 0.80 Source: Zhukauskas (1987). BOOKCOMP, Inc. — John Wiley & Sons / Page 485 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM ARRAYS OF OBJECTS 485 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [485], (47) Lines: 2271 to 2293 ——— -1.06493pt PgVar ——— Long Page PgEnds: T E X [485], (47) TABLE 6.3 Parameter C 2 for Various Tube Rows and Configurations N L 123457101316 Aligned 0.70 0.80 0.86 0.90 0.92 0.95 0.97 0.98 0.99 (Re D,max > 1000) Staggered 0.83 0.87 0.91 0.94 0.95 0.97 0.98 0.98 0.99 (Re D,max 100–1000) Staggered 0.64 0.76 0.84 0.89 0.92 0.95 0.97 0.98 0.99 (Re D,max > 1000) Source: Zhukauskas (1987). It is noted that all properties except Pr s are evaluated at the arithmetic mean of the fluid inlet and outlet temperatures. The constants C and m are listed in Table 6.2, where it can be observed that for S T /S L < 0.7, heat transfer is poor and the aligned tube configuration should not be employed. If N L < 20, a corrected expression can be employed: Nu L N L <20 = C 2 · Nu L N L ≥20 where C 2 = 1.0 for aligned tubes in the range Re D,max and is provided for other conditions in Table 6.3. 6.5.2 Flat Plates Stack of Parallel Plates Consider a stack of parallel plates placed in a free stream (Fig. 6.16). The envelope for the plate stack has a fixed volume, W (width) ×L (length) ×H (height). Morega et al. (1995) conducted numerical analyses on a two-dimensional model where W L, t (the plate thickness), equal to L/20 and q (the heat flux) uniform over the plate surfaces, excluding the edges. The question addressed by Morega et al. is the number of plates needed to maximize the heat transfer performance from the entire plate stack. Fewer plates than the optimum Figure 6.16 Plate stack. (From Morega et al., 1995.) BOOKCOMP, Inc. — John Wiley & Sons / Page 486 / 2nd Proofs / Heat Transfer Handbook / Bejan 486 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [486], (48) Lines: 2293 to 2317 ——— 4.51717pt PgVar ——— Normal Page PgEnds: T E X [486], (48) reduce the amount of heat transfer area in a given volume. A greater population of plates than the optimum increases the resistance to flow through the stack and hence diverts the flow to the free stream zone outside the stack. The heat transfer performance is represented by the nondimensional hot-spot temperature, θ hot = k(T max − T 0 ) q (2nL) (6.160) where T max is the maximum surface temperature on the plate surface, T 0 is the free stream temperature, and k is the fluid thermal conductivity. Figure 6.17 shows θ hot versus the number of plates in the stack, n. The parameter is the Reynolds number Re L = U 0 L/ν, where U 0 is the free stream velocity and ν is the kinematic viscosity of the fluid. Figure 6.17 also shows the curves derived from Nakayama et al. (1988), where the experimental data were obtained with the finned heat sinks having nearly isothermal surfaces. Despite the difference in thermal boundary conditions, the two groups of curves imply a smooth transition of optimum n from low to high Reynolds number regimes. Morega et al. (1995) proposed a relationship for optimum n: n opt 0.26(H/L)Pr 1/4 · Re 1/2 L 1 + 0.26(t/L)Pr 1/4 · Re 1/2 L (6.161) where Pr is the Prandtl number. Equation (6.161) is applicable to cases where Pr ≥ 0.7 and n 1. Figure 6.17 Nondimensional hot spot temperature (θ hot ) versus the number of plates (n). (From Morega et al., 1995.) BOOKCOMP, Inc. — John Wiley & Sons / Page 487 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM ARRAYS OF OBJECTS 487 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [487], (49) Lines: 2317 to 2334 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [487], (49) Offset Strips Offset arrangements of plates or strips (Fig. 6.1d) reproduced with the addition of the symbols in Fig. 6.18 offer the advantage of heat transfer enhance- ment. The boundary layer development is interrupted at the end of the strip and then resumes from the leading edge of the strip in the downstream row. This arrangement allows the boundary layer to be held thin everywhere in the strip array. A similar effect can be obtained with the in-line arrangement of strips, but offsetting the strips from row to row introduces additional favorable effects on heat transfer. The fluid in the offset strips has a longer distance to travel after leaving the trailing edge of a strip to the leading edge of the next strip than in the in-line counterpart. This elongated distance results in a longer elapsed time for the diffusion (or dispersion) of momen- tum and heat. When the fluid velocity is high enough, vortex shedding or turbulence attains a high level in the intervening space between the strips. Thus, the strips after the third row come to be exposed to highly dynamic flow; thermal wakes shed from the upstream strips tend to be diluted by increased level of turbulence so that the indi- vidual strip is washed by a cooler stream thaninthein-linestrip arrangement. Because of these advantages, the offset strip array has been studied by many researchers and used widely in compact heat exchangers. Figure 6.18 includes a table showing the relative strip thicknesses (t/) and the relative strip spacings (s/) studied by DeJong et al. (1998). Those rectangles painted black in the sketch of the strip array are the strips covered with naphthalene to measure the mass transfer rate. The mass transfer data were converted to the heat transfer coefficient using the analogy between mass and heat transfer. The experimental data reveal row-by-row variations of heat transfer coefficient that depend on the Reynolds number. Figure 6.19 shows the friction factor f and Colburn j-factor j plotted as a function of the Reynolds number Re. They are defined as 123 45678Row Flow l s t geometrical parameter experimental geometry numerical geometry tl/ 0.125 0.117 s/l 0.375 0.507 Geometrical parameters for experimental and numerical arrays Figure 6.18 Offset strips. Black strips were naphthalene coated to measure the mass transfer coefficient. The table shows the dimensions. (From DeJong et al., 1998.) BOOKCOMP, Inc. — John Wiley & Sons / Page 488 / 2nd Proofs / Heat Transfer Handbook / Bejan 488 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [488], (50) Lines: 2334 to 2360 ——— 0.34215pt PgVar ——— Normal Page PgEnds: T E X [488], (50) 1 0.1 0.01 0.001 j numerical simulation Joshi and Webb (1987) correlation 100 1000 1,000 Re Figure 6.19 Colburn j-factors and friction factors versus Reynolds number. The curves were based on the Joshi and Webb correlations [eqs. (6.166)–(6.169)]. (From DeJong et al., 1998.) Re = U c d h ν (6.162) where U c is the flow velocity at the minimum free-flow area, d h is the hydraulic diameter, d h = 2(s − t) l + t ν is the fluid kinematic viscosity, and f = 2∆p core ρU 2 c d h 4L core (6.163) where ∆p core is the pressure drop across the entire eight-row test section, L core is the total length of the strip array, ρ is the fluid density, and j = Nu Re Pr m (6.164) where Nu is the Nusselt number, and m = 0.40 [DeJong et al. (1998)] or 0.30 [Joshi and Webb (1987)]. The original j -factor in DeJong et al. (1998) is defined using the BOOKCOMP, Inc. — John Wiley & Sons / Page 489 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM ARRAYS OF OBJECTS 489 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [489], (51) Lines: 2360 to 2413 ——— 5.00323pt PgVar ——— Normal Page PgEnds: T E X [489], (51) Sherwood and Schmidt numbers. For the sake of consistency throughout this section, they are replaced by heat transfer parameters in Nu = d h h av k (6.165) where h av is the average heat transfer coefficient and k is the fluid thermal conduc- tivity. The curves in Fig. 6.19 show the correlations proposed by Joshi and Webb (1987). With the transition Reynolds number denoted as Re * , in the laminar flow range where Re ≤ Re ∗ , f = 8.12Re −0.74 d h −0.41 α −0.02 (6.166) j = 0.53Re −0.50 d h −0.15 α −0.14 (6.167) where α is the aspect ratio; α = s/W, with W taken as the strip width measured normal to the page in Fig. 6.18. In the turbulent flow range where Re ≥ Re ∗ +1000, f = 1.12Re −0.36 d h −0.65 t d h 0.17 (6.168) j = 0.21Re −0.40 d h −0.24 t d h 0.02 (6.169) The transition Reynolds number Re ∗ is given by Re ∗ = Re ∗ b d h b (6.170) with Re ∗ b = 257 s 1.23 t 0.58 (6.171) b = t + 1.328 (Re ) 1/2 (6.172) and Re = U c ν (6.173) The solid symbols in Fig. 6.19 are the results of numerical simulation on the same strip. Figure 6.19 shows the state-of-the-art accuracy in the predictions of f and j . BOOKCOMP, Inc. — John Wiley & Sons / Page 490 / 2nd Proofs / Heat Transfer Handbook / Bejan 490 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [490], (52) Lines: 2413 to 2425 ——— 0.757pt PgVar ——— Short Page PgEnds: T E X [490], (52) 6.6 HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE Figure 6.20 shows a classification of the situations and models encountered in elec- tronics cooling applications: (a) a heated strip that is flush bonded to the substrate surface, (b) a rectangular heat source which is also flush to the substrate surface, (c) an isolated two-dimensional block, (d) a two-dimensional block array, (e) a rectan- gular block and ( f ) an array of rectangular blocks. Ortega et al. (1994) suggest that in many practical situations the substrate is a heat conductor, so that a heat path from the heat source through the substrate to the fluid flow cannot be ignored in what is called conductive/convective conjugate heat transfer. The analytical solution of the Figure 6.20 Configurations of heat sources on substrate. (From Ortega et al., 1994.) BOOKCOMP, Inc. — John Wiley & Sons / Page 491 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM OBJECTS ON A SUBSTRATE 491 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [491], (53) Lines: 2425 to 2457 ——— * 17.97812pt PgVar ——— Short Page PgEnds: T E X [491], (53) conjugate heat transfer problem is difficult, and numerical solutions or experiments have been used in the studies reported in the literature. 6.6.1 Flush-Mounted Heat Sources Heat transfer considerations in flush-mounted heat sources involve less complications than in those for fluid flow. The flow is described by the solution for boundary layer flow or duct flow, particularly where the flow is laminar and conjugate heat transfer is important in laminar flow cases. In turbulent flow, the heat transfer coefficient on the heat source surface is high enough to reduce the heat flow to the substrate to an insignificant level. Gorski and Plumb (1990, 1992) performed numerical analyses on the two-dimensional (Fig. 6.20a) and rectangular (Fig. 6.20b) patch problems. The cross section of the heat source and the substrate is shown in Fig. 6.21. The fluid flow is described by the analytical solution of Blasius for the laminar boundary layer, while the substrate is assumed to be infinitely thick and the numerical solutions are correlated for the two-dimensional strip by Nu = 0.486Pe 0.53 s x s 0.71 k sub k f 0.057 (6.174) where Nu = ¯ h s /k f , ¯ h is the average heat transfer coefficient based on the heat source area (per unit width normal to the page in Fig. 6.21), s is the heat source length, x s is the distance between the leading edge of the substrate and that of the heat source, k sub is the substrate thermal conductivity, and k f is the fluid thermal conductivity. Pe is the P ´ eclet number, Pe = U 0 x s /α, where U 0 is the free stream velocity and α is the fluid thermal diffusivity. Equation (6.174) correlates the numerical solutions within 5% in the parameter ranges 10 3 ≤ Pe ≤ 10 5 , 0.10 ≤ k sub /k f ≤ 10, and 5 ≤ x s / s ≤ 100. For the rectangular patch: Nu = 0.60Pe 0.48 c 2 s 2x s + s 0.63 P s 2A 0.18 k sub k f = 1 (6.175) 0.43Pe 0.52 c 2 s 2x s + s 0.70 P s 2A 0.07 k sub k f = 10 (6.176) Figure 6.21 Two-dimensional flush-mounted heat source. . mass transfer rate. The mass transfer data were converted to the heat transfer coefficient using the analogy between mass and heat transfer. The experimental data reveal row-by-row variations of heat. show similar heat transfer to those in the inner rows. For higher Reynolds numbers, flow turbulence leads to higher heat transfer from inner tubes than from the first row. The heat transfer becomes. difference across which heat is transferred by conduction. With H ν = 1 Pr T ν = κ Pr T y + + δy + 0 (6.150a) then T + = δT + 0 + Pr T κ ln 32.6y + Re k (6.150b) A local heat transfer coefficient